Movie Rating Model and Predictor

Model	Model.Selection	Data
Alpha	Forward Selection	Full model
Beta	Forward Selection	Full model with univariate outliers removed
Gamma	Backward Elimination	Full model
Delta	Backward Elimination	Full model, influential outliers removed

Model Selection

Both forward selection and backward elimination with p-values model selection techniques were used. The forward selection approach optimized adjusted r-squared; whereas the backward elimination method was based upon p-values.

Forward Selection

The forward selection process began with a null model then all variables were added to the model, one-by-one, and the model which provided the greatest improvement over the current best adjusted R-squared was selected. The process repeated with each variable that was not already in the model until all variables were analyzed. Only the models that improved adjusted r-squared were retained at each step.

Backward Elimination

The backward elimination approach began with the full model. A regression analysis was performed and the least significant predictor (that with the highest p-value) was removed from the model. This process repeated, removie only the most least significant predictor at each step, until all predictors had p-values below the present threshold.

Full Model Selection

In the prior section, association and correlation tests were conducted with a 95% confidence level. All categorical variables were significant (Table 2) and were included in the full model. The decision to remove a quantitative variable was based largely upon its correlation with the response variable (Table 3) and its correlations with other quantitative variables (Table 4). Table 5 lists the variables removed and the rationale.

Table 5: Variables excluded from full model

Type	Variable	Description	Rationale
Categorical	actor1	First main actor/actress in the abridged cast of the movie	Not predictive without other data
Categorical	actor2	Second main actor/actress in the abridged cast of the movie	Not predictive without other data
Categorical	actor3	Third main actor/actress in the abridged cast of the movie	Not predictive without other data
Categorical	actor4	Fourth main actor/actress in the abridged cast of the movie	Not predictive without other data
Categorical	actor5	Fifth main actor/actress in the abridged cast of the movie	Not predictive without other data
Categorical	director	Director of the movie	Not predictive without other data
Categorical	dvd_rel_day	Day of the month the movie is released on DVD	No predictive value
Categorical	dvd_rel_month	Month the movie is released on DVD	No predictive value
Categorical	dvd_rel_year	Year the movie is released on DVD	No predictive value
Categorical	imdb_url	Link to IMDB page for the movie	No predictive value
Categorical	rt_url	Link to Rotten Tomatoes page for the movie	No predictive value
Categorical	thtr_rel_date	Date the movie is released in theaters	No predictive value
Categorical	thtr_rel_day	Day of the month the movie is released in theaters	No predictive value
Categorical	thtr_rel_year	Year the movie is released in theaters	No predictive value
Categorical	title	Title of movie	No predictive value
Categorical	title_type	Type of movie (Documentary, Feature Film, TV Movie)	Highly correlated with genre
Numeric	daily_box_office	Daily box office revenue from BoxOfficeMojo.com	Not in data set
Numeric	imdb_num_votes	Number of votes on IMDB	Highly correlated with imdb_num_votes_log
Numeric	votes_per_day	The number of IMDB Votes / thtr_days	Highly correlated with votes_per_day_scores_log
Numeric	votes_per_day_log	Log of votes_per_day	Highly correlated with votes_per_day_scores_log

Thus, the full model is presented in Table 6.

Type	Variable	Description
Categorical	audience_rating	Categorical variable for audience rating on Rotten Tomatoes (Spilled, Upright)
Categorical	best_actor_win	Whether or not one of the main actors in the movie ever won an Oscar (no, yes) – note that this is not necessarily whether the actor won an Oscar for their role in the given movie
Categorical	best_actress_win	Whether or not one of the main actresses in the movie ever won an Oscar (no, yes) – not that this is not necessarily whether the actresses won an Oscar for their role in the given movie
Categorical	best_dir_win	Whether or not the director of the movie ever won an Oscar (no, yes) – not that this is not necessarily whether the director won an Oscar for the given movie
Categorical	best_pic_nom	Whether or not the movie was nominated for a best picture Oscar (no, yes)
Categorical	best_pic_win	Whether or not the movie won a best picture Oscar (no, yes)
Categorical	critics_rating	Categorical variable for critics rating on Rotten Tomatoes (Certified Fresh, Fresh, Rotten)
Categorical	genre	Genre of movie (Action & Adventure, Comedy, Documentary, Drama, Horror, Mystery & Suspense, Other)
Categorical	mpaa_rating	MPAA rating of the movie (G, PG, PG-13, R, Unrated)
Categorical	studio	The studio that produced the film
Categorical	thtr_rel_month	Month the movie is released in theaters
Categorical	thtr_rel_season	Season the movie was released in theaters
Categorical	top200_box	Whether or not the movie is in the Top 200 Box Office list on BoxOfficeMojo (no, yes)
Numeric	audience_score	Audience score on Rotten Tomatoes
Numeric	cast_experience	The sum across all cast members for a film, of the number of films in which each actor appeared
Numeric	cast_experience_log	Log of the sum across all cast members for a film, of the number of films in which each actor appeared
Numeric	cast_votes	Total number of allocated IMDB votes per day for the cast of a film
Numeric	cast_votes_log	Log of cast_votes
Numeric	critics_score	Critics score on Rotten Tomatoes
Numeric	daily_box_office_log	Log of Box office revenue from BoxOfficeMojo.com
Numeric	director_experience	Total number of films in sample for a director
Numeric	director_experience_log	Log of the total number of films directed by the film’s director
Numeric	imdb_num_votes_log	Log number of IMDB votes
Numeric	imdb_rating	Rating on IMDB
Numeric	runtime	Runtime of movie (in minutes)
Numeric	runtime_log	Log runtime of movie (in minutes)
Numeric	thtr_days	Number of days from theatre release date to January 1, 2016
Numeric	thtr_days_log	Log of thtr_days

Model Alpha

For this model, a forward selection procedure was undertaken based upon the full model described above. The variables were added as described in Table 7.

Table 7: Model Alpha forward selection process

Step	Selected	Model.Size	DF	F.statistic	R.Squared	Adjusted.R2	Pct Chg
1	cast_votes_log	1	2 489	536.66	0.52	0.52	0.00
2	genre	2	12 479	56.46	0.56	0.56	6.32
3	critics_score	3	13 478	59.88	0.60	0.59	6.31
4	best_pic_win	4	14 477	57.33	0.61	0.60	1.53
5	cast_experience_log	5	15 476	55.37	0.62	0.61	1.50
6	runtime_log	6	16 475	53.12	0.63	0.62	1.15
7	director_experience_log	7	17 474	50.60	0.63	0.62	0.49
8	thtr_rel_month	8	28 463	30.56	0.64	0.62	0.32
9	best_pic_nom	9	29 462	29.62	0.64	0.62	0.16

Model Overview

This model is defined as follows: \[y_i = (\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5}) + \epsilon_i\]

where:
$y_i$ is the log number of votes for movie $i$
$x_{i1}$ is log of cast_votes for movie $i$
$x_{i2}$ is genre of movie (action & adventure, comedy, documentary, drama, horror, mystery & suspense, other) for movie $i$
$x_{i3}$ is critics score on rotten tomatoes for movie $i$
$x_{i4}$ is whether or not the movie won a best picture oscar (no, yes) for movie $i$
$x_{i5}$ is log of the sum across all cast members for a film, of the number of films in which each actor appeared for movie $i$
$x_{i6}$ is log runtime of movie (in minutes) for movie $i$ $x_{i7}$ is log of the total number of films directed by the film’s director for movie $i$ $x_{i8}$ is month the movie is released in theaters for movie $i$

$\epsilon_i$ is the total residual for the model for movie$i$

As suggested by Figure 1, the model was significant (F(29, 462) = 29.625, p < .001), with an adjusted R-squared of 0.621.

Figure 1 Model Alpha Regression

Analysis of Variance

Figure 2 summarizes the analysis of variance.

Term	Df	Sum Sq	Mean Sq	F Statistic	Pr(>F)	% Var
cast_votes_log	1	1399.493	1399.493	675.751	0.000	52.32
genre	10	110.584	11.058	5.340	0.000	4.13
critics_score	1	96.140	96.140	46.422	0.000	3.59
best_pic_win	1	24.689	24.689	11.921	0.001	0.92
cast_experience_log	1	26.277	26.277	12.688	0.000	0.98
runtime_log	1	18.511	18.511	8.938	0.003	0.69
director_experience_log	1	11.255	11.255	5.435	0.020	0.42
thtr_rel_month	11	26.295	2.390	1.154	0.317	0.98
best_pic_nom	1	4.644	4.644	2.242	0.135	0.17
Residuals	462	956.811	2.071	NA	NA	35.77

Figure 2 Model Alpha analysis of variance

A two-way analysis of variance was conducted on the influence of 9 independent variables on the log number of IMDB votes. The effect of cast_votes_log on the log of IMDB votes presented an F statistic of F(1, 462), = 675.751, p < .001, exhibiting 52.32% of the variance. The significance of genre on the log of IMDB votes presented an F statistic of F(10, 462), = 5.34, p < .001, representing 4.13% of the variance. The influence of critics_score on the log of IMDB votes indicated an F statistic of F(1, 462), = 46.422, p < .001, accounting for 3.59% of the variance. The influence of best_pic_win on the log of IMDB votes presented an F statistic of F(1, 462), = 11.921, p < .001, accounting for 0.92% of the variance. The force of cast_experience_log on the log of IMDB votes indicated an F statistic of F(1, 462), = 12.688, p < .001, exhibiting 0.98% of the variance. The significance of runtime_log on the log of IMDB votes yielded an F statistic of F(1, 462), = 8.938, p < .01, representing 0.69% of the variance. The significance of director_experience_log on the log of IMDB votes yielded an F statistic of F(1, 462), = 5.435, p < .05, representing 0.42% of the variance. The force of thtr_rel_month on the log of IMDB votes presented an F statistic of F(11, 462), = 1.154, p < 0.317, representing 0.98% of the variance. The effect of best_pic_nom on the log of IMDB votes yielded an F statistic of F(1, 462), = 2.242, p < 0.135, accounting for 0.17% of the variance. Finally, residuals exhibited approximately 35.77% of variance.

Model Coefficients

Table 8 Model Alpha Coefficients

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	9.932	2.181	4.553	0.000
cast_votes_log	0.589	0.029	20.021	0.000
genreAnimation	-1.422	0.644	-2.210	0.028
genreArt House & International	-2.225	0.489	-4.551	0.000
genreComedy	-0.976	0.283	-3.453	0.001
genreDocumentary	-2.633	0.368	-7.156	0.000
genreDrama	-1.508	0.243	-6.218	0.000
genreHorror	-0.583	0.418	-1.395	0.164
genreMusical & Performing Arts	-1.929	0.539	-3.579	0.000
genreMystery & Suspense	-1.039	0.318	-3.271	0.001
genreOther	-1.564	0.465	-3.366	0.001
genreScience Fiction & Fantasy	-0.231	0.559	-0.413	0.680
critics_score	0.014	0.003	5.043	0.000
best_pic_winyes	1.128	0.665	1.696	0.091
cast_experience_log	-0.841	0.207	-4.067	0.000
runtime_log	0.676	0.330	2.049	0.041
director_experience_log	0.289	0.117	2.460	0.014
thtr_rel_monthFeb	-0.038	0.371	-0.103	0.918
thtr_rel_monthMar	0.320	0.323	0.991	0.322
thtr_rel_monthApr	-0.043	0.324	-0.132	0.895
thtr_rel_monthMay	-0.033	0.328	-0.101	0.919
thtr_rel_monthJun	0.233	0.289	0.808	0.419
thtr_rel_monthJul	0.712	0.326	2.186	0.029
thtr_rel_monthAug	0.564	0.342	1.650	0.100
thtr_rel_monthSep	-0.024	0.314	-0.076	0.939
thtr_rel_monthOct	0.124	0.294	0.422	0.673
thtr_rel_monthNov	0.311	0.326	0.954	0.341
thtr_rel_monthDec	0.394	0.306	1.289	0.198
best_pic_nomyes	0.644	0.430	1.497	0.135

As shown in Table 8, the predicted log of the number of IMDB votes for a film was 9.932 + a genre factor associated with the genre of the film + plus 0.014 log votes for each point of the critics score 1.128 log votes for log number of films in which the cast appears + -0.841 log votes for each minute of movie runtime. In addition, 0.676 log votes if the fil was nominated for a best picture oscar + 0.289 log votes for each film directed by the director of the film. An additional 0.644 log votes if the film received a best picture oscar. Additional log votes are added (or subtract) based upon the month it was released. The genres, in order of contribution to the log number of IMDB votes estimated, are:

Table 9 Model Alpha Genres and Popularity

Estimate	Std. Error	t value	Pr(>\|t\|)	coef
-0.231	0.559	-0.413	0.680	genreScience Fiction & Fantasy
-0.583	0.418	-1.395	0.164	genreHorror
-0.976	0.283	-3.453	0.001	genreComedy
-1.039	0.318	-3.271	0.001	genreMystery & Suspense
-1.422	0.644	-2.210	0.028	genreAnimation
-1.508	0.243	-6.218	0.000	genreDrama
-1.564	0.465	-3.366	0.001	genreOther
-1.929	0.539	-3.579	0.000	genreMusical & Performing Arts
-2.225	0.489	-4.551	0.000	genreArt House & International
-2.633	0.368	-7.156	0.000	genreDocumentary

According to this model, Science Fiction and Fantasy, Horror and Comedy films are the most popular.

Table 10 Model Alpha Timing and Popularity

Estimate	Std. Error	t value	Pr(>\|t\|)	coef
0.712	0.326	2.186	0.029	thtr_rel_monthJul
0.564	0.342	1.650	0.100	thtr_rel_monthAug
0.394	0.306	1.289	0.198	thtr_rel_monthDec
0.320	0.323	0.991	0.322	thtr_rel_monthMar
0.311	0.326	0.954	0.341	thtr_rel_monthNov
0.233	0.289	0.808	0.419	thtr_rel_monthJun
0.124	0.294	0.422	0.673	thtr_rel_monthOct
-0.024	0.314	-0.076	0.939	thtr_rel_monthSep
-0.033	0.328	-0.101	0.919	thtr_rel_monthMay
-0.038	0.371	-0.103	0.918	thtr_rel_monthFeb
-0.043	0.324	-0.132	0.895	thtr_rel_monthApr

As indicated in Table 10, mid to late summer and November are the best months in which to launch a film.

Model Diagnostics

Linearity

The linearity of each predictor with the log number of IMDB votes is illustrated in Figure 3.

Figure 3 Model Alpha linearity plots

A review of the partial scatterplots indicated that linearity was a reasonable assumption for this model (despite the presence of several influential points). A linear hypothesis test was conducted to test the linearity assumption. The results were significant (F(29), p < .001). As such, the linearity assumption was met in this case.

Homoscedasticity

The following plot (Figure 4) of the residuals versus the fitted values provides a graphic indication of the distribution of residual variances. Figure 4 Model Alpha homoscedasticity plot

The residuals plot above indicated equal dispersion of residuals about zero mean. A Breuschâ€“Pagan test test was conducted to test the homoscedasticity assumption. The results were significant (F(1), p < .01). As such, the homoscedasticity assumption was met in this case.

Residuals

The histogram and the normal Q-Q plot in Figure 5 illustrate the distribution of residuals.

Figure 5 Model Alpha residuals plot

The histogram and normal Q-Q plot suggested a normal distribution of residuals. A review of the Shapiroi-Wilk test (SW = 0.986, p = 0) and the skewness (-0.463) and kurtosis (3.194) supported the assumption of normaility.

Multicollinearity

As shown in Figure 6 and Table 11, collinearity did not appear extant for this model. Variance inflation factors were computed for each predictor in the model. The maximum VIF of 2 did not exceed the threshold of 4. As such, the absense of multicollinearity was assumed for this model.
Figure 6: Model Alpha correlations among quantitative predictors

Table 11 Model Alpha variance inflation Factors

	GVIF	Df	GVIF^(1/(2*Df))
cast_votes_log	1.612	1	1.270
genre	2.139	10	1.039
critics_score	1.365	1	1.168
best_pic_win	1.475	1	1.215
cast_experience_log	1.632	1	1.278
runtime_log	1.427	1	1.195
director_experience_log	1.160	1	1.077
thtr_rel_month	1.532	11	1.020
best_pic_nom	1.549	1	1.245

Outliers

Figure 7 Model Alpha Outliers

Examination of the residuals versus leverage plot and case-wise diagnostics such as Cook’s distance revealed 22 cases exerting undue influence on the model. The discern the effect of these outliers on the model, a new model (Model B) was created without the outliers removed.

Model Beta

For this model, a forward selection procedure was undertaken based upon the full model with outliers from Model Alpha removed. The variables were added as described in Table 12

Table 12: Model Beta forward selection process

Step	Selected	Model.Size	DF	F.statistic	R.Squared	Adjusted.R2	Pct Chg
1	cast_votes_log	1	2 467	651.36	0.58	0.58	0.00
2	genre	2	12 457	67.79	0.62	0.61	4.98
3	critics_score	3	13 456	71.52	0.65	0.64	5.40
4	cast_experience_log	4	14 455	68.77	0.66	0.65	1.40
5	best_pic_win	5	15 454	66.67	0.67	0.66	1.53
6	runtime_log	6	16 453	64.20	0.68	0.67	1.06
7	director_experience_log	7	17 452	61.51	0.68	0.67	0.60
8	mpaa_rating	8	21 448	49.56	0.69	0.68	0.15
9	thtr_rel_month	9	32 437	32.50	0.70	0.68	0.15
10	best_actress_win	10	33 436	31.62	0.70	0.68	0.15

Model Overview

This model is defined as follows: \[y_i = (\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5}+ \beta_5 x_{i6}+ \beta_5 x_{i7}+ \beta_5 x_{i8}+ \beta_5 x_{i9}+ \beta_5 x_{i10}) + \epsilon_i\]

where:
$y_i$ is the log number of votes for movie $i$
$x_{i1}$ is log of cast_votes for movie $i$
$x_{i2}$ is genre of movie (action & adventure, comedy, documentary, drama, horror, mystery & suspense, other) for movie $i$
$x_{i3}$ is critics score on rotten tomatoes for movie $i$
$x_{i4}$ is log of the sum across all cast members for a film, of the number of films in which each actor appeared for movie $i$
$x_{i5}$ is whether or not the movie won a best picture oscar (no, yes) for movie $i$
$x_{i6}$ is log runtime of movie (in minutes) for movie $i$
$x_{i7}$ is log of the total number of films directed by the film’s director for movie $i$
$x_{i8}$ is mpaa rating of the movie (g, pg, pg-13, r, unrated) for movie $i$

$\epsilon_i$ is the total residual for the model for movie$i$

As suggested by Figure 8, the model was significant (F(33, 436) = 31.618, p < .001), with an adjusted R-squared of 0.677.

Figure 8 Model Beta Regression

Analysis of Variance

Figure 9 summarizes the analysis of variance.

Term	Df	Sum Sq	Mean Sq	F Statistic	Pr(>F)	% Var
cast_votes_log	1	1411.992	1411.992	843.213	0.000	58.24
genre	10	91.169	9.117	5.444	0.000	3.76
critics_score	1	80.036	80.036	47.796	0.000	3.30
cast_experience_log	1	23.489	23.489	14.027	0.000	0.97
best_pic_win	1	24.312	24.312	14.519	0.000	1.00
runtime_log	1	17.794	17.794	10.626	0.001	0.73
director_experience_log	1	12.500	12.500	7.464	0.007	0.52
mpaa_rating	4	8.385	2.096	1.252	0.288	0.35
thtr_rel_month	11	21.177	1.925	1.150	0.321	0.87
best_actress_win	1	3.390	3.390	2.025	0.155	0.14
Residuals	436	730.099	1.675	NA	NA	30.12

Figure 9 Model Beta analysis of variance

A two-way analysis of variance was conducted on the influence of 10 independent variables on the log number of IMDB votes. The influence of cast_votes_log on the log of IMDB votes yielded an F statistic of F(1, 436), = 843.213, p < .001, accounting for 58.24% of the variance. The force of genre on the log of IMDB votes produced an F statistic of F(10, 436), = 5.444, p < .001, exhibiting 3.76% of the variance. The significance of critics_score on the log of IMDB votes yielded an F statistic of F(1, 436), = 47.796, p < .001, exhibiting 3.3% of the variance. The effect of cast_experience_log on the log of IMDB votes yielded an F statistic of F(1, 436), = 14.027, p < .001, exhibiting 0.97% of the variance. The significance of best_pic_win on the log of IMDB votes indicated an F statistic of F(1, 436), = 14.519, p < .001, accounting for 1% of the variance. The force of runtime_log on the log of IMDB votes produced an F statistic of F(1, 436), = 10.626, p < .01, accounting for 0.73% of the variance. The force of director_experience_log on the log of IMDB votes yielded an F statistic of F(1, 436), = 7.464, p < .01, accounting for 0.52% of the variance. The force of mpaa_rating on the log of IMDB votes presented an F statistic of F(4, 436), = 1.252, p < 0.288, representing 0.35% of the variance. The influence of thtr_rel_month on the log of IMDB votes yielded an F statistic of F(11, 436), = 1.15, p < 0.321, representing 0.87% of the variance. The force of best_actress_win on the log of IMDB votes produced an F statistic of F(1, 436), = 2.025, p < 0.155, representing 0.14% of the variance. Finally, residuals exhibited some 30.12% of variance.

Model Coefficients

Table 13 Model Beta Coefficients

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	8.911	2.043	4.361	0.000
cast_votes_log	0.609	0.028	21.379	0.000
genreAnimation	-1.351	0.742	-1.820	0.069
genreArt House & International	-1.543	0.522	-2.955	0.003
genreComedy	-0.800	0.260	-3.077	0.002
genreDocumentary	-2.145	0.376	-5.710	0.000
genreDrama	-1.368	0.224	-6.097	0.000
genreHorror	-0.384	0.384	-0.999	0.318
genreMusical & Performing Arts	-1.807	0.539	-3.351	0.001
genreMystery & Suspense	-0.875	0.297	-2.949	0.003
genreOther	-1.427	0.488	-2.922	0.004
genreScience Fiction & Fantasy	-0.159	0.504	-0.316	0.752
critics_score	0.014	0.003	5.329	0.000
cast_experience_log	-0.788	0.193	-4.073	0.000
best_pic_winyes	1.665	0.523	3.185	0.002
runtime_log	0.825	0.308	2.679	0.008
director_experience_log	0.260	0.107	2.425	0.016
mpaa_ratingPG	-0.277	0.429	-0.645	0.519
mpaa_ratingPG-13	-0.041	0.440	-0.092	0.927
mpaa_ratingR	-0.273	0.420	-0.649	0.517
mpaa_ratingUnrated	-0.756	0.511	-1.480	0.140
thtr_rel_monthFeb	0.065	0.351	0.184	0.854
thtr_rel_monthMar	0.424	0.302	1.403	0.161
thtr_rel_monthApr	0.048	0.299	0.161	0.872
thtr_rel_monthMay	0.183	0.308	0.595	0.552
thtr_rel_monthJun	0.199	0.267	0.747	0.455
thtr_rel_monthJul	0.599	0.297	2.017	0.044
thtr_rel_monthAug	0.699	0.315	2.217	0.027
thtr_rel_monthSep	-0.049	0.291	-0.168	0.866
thtr_rel_monthOct	0.079	0.267	0.297	0.767
thtr_rel_monthNov	0.291	0.300	0.970	0.333
thtr_rel_monthDec	0.269	0.278	0.965	0.335
best_actress_winyes	-0.281	0.198	-1.423	0.155

As shown in Table 13, the predicted log of the number of IMDB votes for a film was 8.911 + 0.609 log votes for each log vote earned by a cast member. A factor is added for the various genres as follows:

Table 14 Model Beta Genres and Popularity

Estimate	Std. Error	t value	Pr(>\|t\|)	coef
-0.159	0.504	-0.316	0.752	genreScience Fiction & Fantasy
-0.384	0.384	-0.999	0.318	genreHorror
-0.800	0.260	-3.077	0.002	genreComedy
-0.875	0.297	-2.949	0.003	genreMystery & Suspense
-1.351	0.742	-1.820	0.069	genreAnimation
-1.368	0.224	-6.097	0.000	genreDrama
-1.427	0.488	-2.922	0.004	genreOther
-1.543	0.522	-2.955	0.003	genreArt House & International
-1.807	0.539	-3.351	0.001	genreMusical & Performing Arts
-2.145	0.376	-5.710	0.000	genreDocumentary

It should be noted that most, not all estimates were statistically significant.

An additional 0.014 log votes are earned for each point of the critics score -0.788 log votes for log number of the sum of films in which the top 5 cast members appeared, + 1.665 log votes for each log minute of runtime, plus 1.665 log votes if the film won best picture, + 0.26 log votes for the log of the number of films that the director had directed, + -0.281 log votes if the film was nominated for best picture. The estimate of the log number of IMDB votes is adjusted according to the month of release as follows:

Table 15 Model Beta Timing and Popularity

Estimate	Std. Error	t value	Pr(>\|t\|)	coef
0.699	0.315	2.217	0.027	thtr_rel_monthAug
0.599	0.297	2.017	0.044	thtr_rel_monthJul
0.424	0.302	1.403	0.161	thtr_rel_monthMar
0.291	0.300	0.970	0.333	thtr_rel_monthNov
0.269	0.278	0.965	0.335	thtr_rel_monthDec
0.199	0.267	0.747	0.455	thtr_rel_monthJun
0.183	0.308	0.595	0.552	thtr_rel_monthMay
0.079	0.267	0.297	0.767	thtr_rel_monthOct
0.065	0.351	0.184	0.854	thtr_rel_monthFeb
0.048	0.299	0.161	0.872	thtr_rel_monthApr
-0.049	0.291	-0.168	0.866	thtr_rel_monthSep

In should be known; however, that many of the estimates for the months were not significant.

Lastly, an adjustment to the estimate is made for the MPAA rating.

Model Diagnostics

Linearity

The linearity of each predictor with the log number of IMDB votes is illustrated in Figure 10.

Figure 10 Model beta linearity plots

A review of the partial scatterplots indicated that linearity was a reasonable assumption for this model (despite the presence of several influential points). A linear hypothesis test was conducted to test the linearity assumption. The results were significant (F(33), p < .001). As such, the linearity assumption was met in this case.

Homoscedasticity

The following plot (Figure 11) of the residuals versus the fitted values provides a graphic indication of the distribution of residual variances. Figure 11 Model beta homoscedasticity plot

The residuals plot above indicated equal dispersion of residuals about zero mean. A Breuschâ€“Pagan test test was conducted to test the homoscedasticity assumption. The results were significant (F(1), p < .05). As such, the homoscedasticity assumption was met in this case.

Residuals

The histogram and the normal Q-Q plot in Figure 12 illustrate the distribution of residuals.

Figure 12 Model beta residuals plot

The histogram and normal Q-Q plot suggested a normal distribution of residuals. A review of the Shapiroi-Wilk test (SW = 0.991, p = 0.007) and the skewness (-0.297) and kurtosis (2.842) supported the assumption of normaility.

Multicollinearity

As shown in Figure 13 and Table 16, collinearity appeared extant for this moiiidel. Variance inflation factors were computed for each predictor in the model. The maximum VIF of 4 exceeded the threshold of 4. As such, the correlation among the predictors would require further consideration. The multicollinearity assumption was not met for this model.
Figure 13: Correlations among quantitative predictors

Table 16 Model Beta Variance Inflation Factors

	GVIF	Df	GVIF^(1/(2*Df))
cast_votes_log	1.752	1	1.324
genre	4.060	10	1.073
critics_score	1.435	1	1.198
cast_experience_log	1.707	1	1.306
best_pic_win	1.126	1	1.061
runtime_log	1.488	1	1.220
director_experience_log	1.160	1	1.077
mpaa_rating	2.835	4	1.139
thtr_rel_month	1.658	11	1.023
best_actress_win	1.205	1	1.098

Outliers

Figure 14 Model Beta Outliers

Examination of the residuals versus leverage plot and case-wise diagnostics such as Cook’s distance revealed 19 cases exerting undue influence on the model.

Model Gamma

For this model, a backward elimination procedure was undertaken based upon the full model The variables were removed as described in Table 17

Table 17: Model Gamma

Steps	Removed	p.value
1	best_actress_win	0.53

Model Overview

This model is defined as follows: \[y_i = (\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5} + \beta_1 x_{i6} + \beta_2 x_{i7} + \beta_3 x_{i8} + \beta_4 x_{i9} + \beta_5 x_{i10} + \beta_5 x_{i11} + \beta_5 x_{i12} + \beta_5 x_{i13}) + \epsilon_i\]

where:
$y_i$ is the log number of votes for movie $i$
$x_{i1}$ is genre of movie (action & adventure, comedy, documentary, drama, horror, mystery & suspense, other) for movie $i$ $x_{i2}$ is mpaa rating of the movie (g, pg, pg-13, r, unrated) for movie $i$ $x_{i3}$ is month the movie is released in theaters for movie $i$ $x_{i4}$ is whether or not the movie was nominated for a best picture oscar (no, yes) for movie $i$ $x_{i5}$ is whether or not the movie won a best picture oscar (no, yes) for movie $i$ $x_{i6}$ is whether or not one of the main actors in the movie ever won an oscar (no, yes) – note that this is not necessarily whether the actor won an oscar for their role in the given movie for movie $i$ $x_{i7}$ is whether or not the director of the movie ever won an oscar (no, yes) – not that this is not necessarily whether the director won an oscar for the given movie for movie $i$ $x_{i8}$ is critics score on rotten tomatoes for movie $i$ $x_{i9}$ is number of days from theatre release date to january 1, 2,016 for movie $i$ $x_{i10}$ is log runtime of movie (in minutes) for movie $i$ $x_{i11}$ is log of cast_votes for movie $i$ $x_{i12}$ is log of the total number of films directed by the film’s director for movie $i$ $x_{i13}$ is log of the sum across all cast members for a film, of the number of films in which each actor appeared for movie $i$ $\epsilon_i$ is the total residual for the model for movie$i$

As suggested by Figure 15, the model was significant (F(36, 455) = 23.956, p < .001), with an adjusted R-squared of 0.621.

Figure 15 Model Gamma Regression

Analysis of Variance

Figure 16 summarizes the analysis of variance.

Term	Df	Sum Sq	Mean Sq	F Statistic	Pr(>F)	% Var
genre	10	376.261	37.626	18.196	0.000	14.07
mpaa_rating	4	99.942	24.986	12.083	0.000	3.74
thtr_rel_month	11	103.141	9.376	4.534	0.000	3.86
best_pic_nom	1	111.151	111.151	53.752	0.000	4.16
best_pic_win	1	26.967	26.967	13.041	0.000	1.01
best_actor_win	1	17.770	17.770	8.594	0.004	0.66
best_dir_win	1	25.382	25.382	12.275	0.001	0.95
critics_score	1	145.037	145.037	70.139	0.000	5.42
thtr_days	1	207.620	207.620	100.404	0.000	7.76
runtime_log	1	60.489	60.489	29.252	0.000	2.26
cast_votes_log	1	519.590	519.590	251.271	0.000	19.43
director_experience_log	1	9.104	9.104	4.403	0.036	0.34
cast_experience_log	1	31.373	31.373	15.172	0.000	1.17
Residuals	455	940.870	2.068	NA	NA	35.18

Figure 16 Model Gamma analysis of variance

A two-way analysis of variance was conducted on the influence of 13 independent variables on the log number of IMDB votes. The effect of genre on the log of IMDB votes indicated an F statistic of F(10, 455), = 18.196, p < .001, accounting for 14.07% of the variance. The force of mpaa_rating on the log of IMDB votes indicated an F statistic of F(4, 455), = 12.083, p < .001, expressing 3.74% of the variance. The influence of thtr_rel_month on the log of IMDB votes presented an F statistic of F(11, 455), = 4.534, p < .001, expressing 3.86% of the variance. The force of best_pic_nom on the log of IMDB votes presented an F statistic of F(1, 455), = 53.752, p < .001, exhibiting 4.16% of the variance. The force of best_pic_win on the log of IMDB votes indicated an F statistic of F(1, 455), = 13.041, p < .001, accounting for 1.01% of the variance. The significance of best_actor_win on the log of IMDB votes indicated an F statistic of F(1, 455), = 8.594, p < .01, representing 0.66% of the variance. The force of best_dir_win on the log of IMDB votes yielded an F statistic of F(1, 455), = 12.275, p < .001, representing 0.95% of the variance. The force of critics_score on the log of IMDB votes yielded an F statistic of F(1, 455), = 70.139, p < .001, expressing 5.42% of the variance. The significance of thtr_days on the log of IMDB votes presented an F statistic of F(1, 455), = 100.404, p < .001, representing 7.76% of the variance. The influence of runtime_log on the log of IMDB votes indicated an F statistic of F(1, 455), = 29.252, p < .001, expressing 2.26% of the variance. The effect of cast_votes_log on the log of IMDB votes presented an F statistic of F(1, 455), = 251.271, p < .001, expressing 19.43% of the variance. The force of director_experience_log on the log of IMDB votes indicated an F statistic of F(1, 455), = 4.403, p < .05, accounting for 0.34% of the variance. The significance of cast_experience_log on the log of IMDB votes indicated an F statistic of F(1, 455), = 15.172, p < .001, exhibiting 1.17% of the variance. Finally, residuals expressed a 35.18% of variance.

Model Coefficients

Table 18 Model Gamma Coefficients

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	10.042	2.284	4.398	0.000
genreAnimation	-1.432	0.704	-2.036	0.042
genreArt House & International	-2.088	0.501	-4.171	0.000
genreComedy	-1.011	0.286	-3.535	0.000
genreDocumentary	-2.484	0.411	-6.042	0.000
genreDrama	-1.492	0.249	-6.006	0.000
genreHorror	-0.466	0.426	-1.092	0.276
genreMusical & Performing Arts	-1.954	0.545	-3.588	0.000
genreMystery & Suspense	-0.989	0.322	-3.072	0.002
genreOther	-1.474	0.471	-3.132	0.002
genreScience Fiction & Fantasy	-0.278	0.560	-0.496	0.620
mpaa_ratingPG	-0.129	0.461	-0.280	0.780
mpaa_ratingPG-13	0.135	0.483	0.279	0.780
mpaa_ratingR	-0.154	0.459	-0.335	0.738
mpaa_ratingUnrated	-0.601	0.566	-1.061	0.289
thtr_rel_monthFeb	-0.015	0.371	-0.041	0.967
thtr_rel_monthMar	0.387	0.325	1.190	0.235
thtr_rel_monthApr	-0.020	0.329	-0.059	0.953
thtr_rel_monthMay	0.019	0.331	0.058	0.953
thtr_rel_monthJun	0.288	0.293	0.983	0.326
thtr_rel_monthJul	0.725	0.329	2.204	0.028
thtr_rel_monthAug	0.534	0.346	1.544	0.123
thtr_rel_monthSep	0.075	0.320	0.235	0.814
thtr_rel_monthOct	0.157	0.299	0.525	0.600
thtr_rel_monthNov	0.388	0.331	1.172	0.242
thtr_rel_monthDec	0.434	0.307	1.412	0.159
best_pic_nomyes	0.701	0.436	1.606	0.109
best_pic_winyes	0.843	0.700	1.205	0.229
best_actor_winyes	-0.194	0.209	-0.931	0.352
best_dir_winyes	0.429	0.289	1.485	0.138
critics_score	0.015	0.003	5.205	0.000
thtr_days	0.000	0.000	-0.600	0.549
runtime_log	0.680	0.348	1.956	0.051
cast_votes_log	0.569	0.037	15.535	0.000
director_experience_log	0.249	0.125	1.995	0.047
cast_experience_log	-0.834	0.214	-3.895	0.000

The coefficients for this model were the same of those for model Beta; however, the order in which the variables were added was different as was the estimates.

Model Diagnostics

Linearity

The linearity of each predictor with the log number of IMDB votes is illustrated in Figure 17.

Figure 17 Model beta linearity plots

A review of the partial scatterplots indicated that linearity was a reasonable assumption for this model (despite the presence of several influential points). A linear hypothesis test was conducted to test the linearity assumption. The results were significant (F(36), p < .001). As such, the linearity assumption was met in this case.

Homoscedasticity

The following plot (Figure 18) of the residuals versus the fitted values provides a graphic indication of the distribution of residual variances. Figure 18 Model beta homoscedasticity plot

The residuals plot above indicated equal dispersion of residuals about zero mean. A Breuschâ€“Pagan test test was conducted to test the homoscedasticity assumption. The results were significant (F(1), p < .01). As such, the homoscedasticity assumption was met in this case.

Residuals

The histogram and the normal Q-Q plot in Figure 19 illustrate the distribution of residuals.

Figure 19 Model beta residuals plot

The histogram and normal Q-Q plot suggested a normal distribution of residuals. A review of the Shapiroi-Wilk test (SW = 0.99, p = 0.001) and the skewness (-0.386) and kurtosis (3.139) supported the assumption of normaility.

Multicollinearity

As shown in Figure 20 and Table 19, collinearity appeared extant for this moiiidel. Variance inflation factors were computed for each predictor in the model. The maximum VIF of 4 exceeded the threshold of 4. As such, the correlation among the predictors would require further consideration. The multicollinearity assumption was not met for this model.
Figure 20: Correlations among quantitative predictors

Table 19 Model Gamma Variance Inflation Factors

	GVIF	Df	GVIF^(1/(2*Df))
genre	4.405	10	1.077
mpaa_rating	3.173	4	1.155
thtr_rel_month	1.895	11	1.029
best_pic_nom	1.595	1	1.263
best_pic_win	1.636	1	1.279
best_actor_win	1.232	1	1.110
best_dir_win	1.381	1	1.175
critics_score	1.499	1	1.224
thtr_days	1.966	1	1.402
runtime_log	1.589	1	1.260
cast_votes_log	2.498	1	1.580
director_experience_log	1.313	1	1.146
cast_experience_log	1.755	1	1.325

Outliers

Figure 21 Model Gamma Outliers

Examination of the residuals versus leverage plot and case-wise diagnostics such as Cook’s distance revealed 20 cases exerting undue influence on the model. To discern the effect of the influential points on the model, a new model (Model Delta) was created without the influential points of this model.

Model Delta

For this model, a backward elimination procedure was undertaken based upon the full model The variables were removed as described in Table 20

Table 20: Model Delta

Steps	Removed	p.value
1	best_actress_win	0.52

Model Overview

This model is defined as follows: \[y_i = (\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5} + \beta_1 x_{i6} + \beta_2 x_{i7} + \beta_3 x_{i8} + \beta_4 x_{i9} + \beta_5 x_{i10}) + \beta_5 x_{i11}+ \beta_5 x_{i12}) + \epsilon_i\]

where:
$\epsilon_i$ is the total residual for the model for movie$i$

As suggested by Figure 22, the model was significant (F(36, 435) = 28.66, p < .001), with an adjusted R-squared of 0.673.

Figure 22 Model Delta Regression

Analysis of Variance

Figure 23 summarizes the analysis of variance.

Term	Df	Sum Sq	Mean Sq	F Statistic	Pr(>F)	% Var
genre	10	368.668	36.867	21.669	0.000	15.07
mpaa_rating	4	113.484	28.371	16.676	0.000	4.64
thtr_rel_month	11	89.408	8.128	4.777	0.000	3.65
best_pic_nom	1	99.674	99.674	58.586	0.000	4.07
best_pic_win	1	31.941	31.941	18.774	0.000	1.31
best_actor_win	1	21.859	21.859	12.848	0.000	0.89
best_dir_win	1	23.668	23.668	13.911	0.000	0.97
critics_score	1	122.917	122.917	72.248	0.000	5.02
thtr_days	1	200.765	200.765	118.005	0.000	8.21
runtime_log	1	72.442	72.442	42.580	0.000	2.96
cast_votes_log	1	516.605	516.605	303.649	0.000	21.11
director_experience_log	1	11.883	11.883	6.985	0.009	0.49
cast_experience_log	1	33.313	33.313	19.580	0.000	1.36
Residuals	435	740.075	1.701	NA	NA	30.25

Figure 23 Model Delta analysis of variance

A two-way analysis of variance was conducted on the influence of 13 independent variables on the log number of IMDB votes. The effect of genre on the log of IMDB votes presented an F statistic of F(10, 435), = 21.669, p < .001, expressing 15.07% of the variance. The influence of mpaa_rating on the log of IMDB votes presented an F statistic of F(4, 435), = 16.676, p < .001, expressing 4.64% of the variance. The effect of thtr_rel_month on the log of IMDB votes presented an F statistic of F(11, 435), = 4.777, p < .001, accounting for 3.65% of the variance. The force of best_pic_nom on the log of IMDB votes yielded an F statistic of F(1, 435), = 58.586, p < .001, representing 4.07% of the variance. The force of best_pic_win on the log of IMDB votes presented an F statistic of F(1, 435), = 18.774, p < .001, expressing 1.31% of the variance. The significance of best_actor_win on the log of IMDB votes yielded an F statistic of F(1, 435), = 12.848, p < .001, expressing 0.89% of the variance. The force of best_dir_win on the log of IMDB votes indicated an F statistic of F(1, 435), = 13.911, p < .001, exhibiting 0.97% of the variance. The influence of critics_score on the log of IMDB votes yielded an F statistic of F(1, 435), = 72.248, p < .001, expressing 5.02% of the variance. The force of thtr_days on the log of IMDB votes yielded an F statistic of F(1, 435), = 118.005, p < .001, expressing 8.21% of the variance. The effect of runtime_log on the log of IMDB votes presented an F statistic of F(1, 435), = 42.58, p < .001, accounting for 2.96% of the variance. The force of cast_votes_log on the log of IMDB votes yielded an F statistic of F(1, 435), = 303.649, p < .001, accounting for 21.11% of the variance. The effect of director_experience_log on the log of IMDB votes yielded an F statistic of F(1, 435), = 6.985, p < .01, representing 0.49% of the variance. The influence of cast_experience_log on the log of IMDB votes produced an F statistic of F(1, 435), = 19.58, p < .001, representing 1.36% of the variance. Finally, residuals represented some 30.25% of variance.

Model Coefficients

Table 21 Model Delta Coefficients

	Estimate	Std. Error	t value	Pr(>\|t\|)
(Intercept)	9.669	2.120	4.562	0.000
genreAnimation	-1.144	0.763	-1.499	0.135
genreArt House & International	-1.735	0.475	-3.654	0.000
genreComedy	-0.914	0.263	-3.479	0.001
genreDocumentary	-2.303	0.388	-5.941	0.000
genreDrama	-1.452	0.227	-6.399	0.000
genreHorror	-0.433	0.388	-1.116	0.265
genreMusical & Performing Arts	-1.864	0.546	-3.415	0.001
genreMystery & Suspense	-0.883	0.295	-2.989	0.003
genreOther	-1.749	0.472	-3.705	0.000
genreScience Fiction & Fantasy	-0.224	0.509	-0.440	0.660
mpaa_ratingPG	-0.042	0.454	-0.092	0.926
mpaa_ratingPG-13	0.231	0.472	0.490	0.624
mpaa_ratingR	-0.033	0.451	-0.073	0.942
mpaa_ratingUnrated	-0.545	0.548	-0.995	0.320
thtr_rel_monthFeb	-0.033	0.352	-0.095	0.925
thtr_rel_monthMar	0.458	0.302	1.515	0.130
thtr_rel_monthApr	-0.093	0.302	-0.309	0.758
thtr_rel_monthMay	0.217	0.312	0.695	0.487
thtr_rel_monthJun	0.096	0.270	0.356	0.722
thtr_rel_monthJul	0.586	0.301	1.951	0.052
thtr_rel_monthAug	0.643	0.322	1.997	0.046
thtr_rel_monthSep	-0.045	0.297	-0.152	0.879
thtr_rel_monthOct	0.037	0.274	0.137	0.891
thtr_rel_monthNov	0.166	0.304	0.547	0.585
thtr_rel_monthDec	0.233	0.282	0.827	0.409
best_pic_nomyes	0.500	0.412	1.216	0.225
best_pic_winyes	1.055	0.644	1.639	0.102
best_actor_winyes	-0.103	0.195	-0.530	0.596
best_dir_winyes	0.350	0.263	1.328	0.185
critics_score	0.013	0.003	4.919	0.000
thtr_days	0.000	0.000	-0.241	0.809
runtime_log	0.741	0.322	2.298	0.022
cast_votes_log	0.593	0.035	17.057	0.000
director_experience_log	0.289	0.115	2.515	0.012
cast_experience_log	-0.882	0.199	-4.425	0.000

The coefficients foer this model was the same as those of model Gamma. The order was different as were the estimates.

Model Diagnostics

Linearity

The linearity of each predictor with the log number of IMDB votes is illustrated in Figure 24.

Figure 24 Model beta linearity plots

A review of the partial scatterplots indicated that linearity was a reasonable assumption for this model (despite the presence of several influential points). A linear hypothesis test was conducted to test the linearity assumption. The results were significant (F(36), p < .001). As such, the linearity assumption was met in this case.

Homoscedasticity

The following plot (Figure 25) of the residuals versus the fitted values provides a graphic indication of the distribution of residual variances. Figure 25 Model beta homoscedasticity plot

The residuals plot above indicated equal dispersion of residuals about zero mean. A Breuschâ€“Pagan test test was conducted to test the homoscedasticity assumption. The results were significant (F(1), p < .05). As such, the homoscedasticity assumption was met in this case.

Residuals

The histogram and the normal Q-Q plot in Figure 26 illustrate the distribution of residuals.

Figure 26 Model beta residuals plot

The histogram and normal Q-Q plot suggested a normal distribution of residuals. A review of the Shapiroi-Wilk test (SW = 0.992, p = 0.011) and the skewness (-0.281) and kurtosis (2.851) supported the assumption of normaility.

Multicollinearity

As shown in Figure 27 and Table 22, collinearity appeared extant for this moiiidel. Variance inflation factors were computed for each predictor in the model. The maximum VIF of 4 exceeded the threshold of 4. As such, the correlation among the predictors would require further consideration. The multicollinearity assumption was not met for this model.
Figure 27: Correlations among quantitative predictors

Table 22 Model Delta Variance Inflation Factors

	GVIF	Df	GVIF^(1/(2*Df))
genre	4.481	10	1.078
mpaa_rating	3.226	4	1.158
thtr_rel_month	1.886	11	1.029
best_pic_nom	1.631	1	1.277
best_pic_win	1.679	1	1.296
best_actor_win	1.234	1	1.111
best_dir_win	1.388	1	1.178
critics_score	1.502	1	1.226
thtr_days	2.055	1	1.433
runtime_log	1.602	1	1.266
cast_votes_log	2.599	1	1.612
director_experience_log	1.318	1	1.148
cast_experience_log	1.759	1	1.326

Outliers

Figure 28 Model Delta Outliers

Examination of the residuals versus leverage plot and case-wise diagnostics such as Cook’s distance revealed 19 cases exerting undue influence on the model.

Model Comparisons

To summarize, models Alpha and Beta were constructed using forward selection and models Gamma and Delta were developed via backward elimination. Models Beta and Delta were fitted without the influential data points from models Alpha and Gamma respectively.

Table 23 Summary of models

Model	Size	df	df Residuals	F Statistic	RMSE	Residual SE	R-Squared	Adj R-Squared	% Variance
Model Alpha	9	29	462	29.625	1.439	1.439	0.642	0.621	64.227
Model Beta	10	33	436	31.618	1.294	1.294	0.699	0.677	69.885
Model Gamma	13	36	455	23.956	1.438	1.438	0.648	0.621	64.823
Model Delta	13	36	435	28.660	1.304	1.304	0.698	0.673	69.752

Forward Selection vs. Backward Elimination

As shown in Table 23, the forward selection algorithm produced fewer predictors than the backward elimination algorithm. Notwithstanding, the differences in root mean square error for the models was not significant -0.08% and -0.79%. Similarly, the differences in adjusted R-squared were -0.09% and -0.53%, not a significant difference. Lastly the differences in the percent variance explained by the models also lacking in significance (0.93% and -0.19%).

Drop or Not

The Beta and Delta models were trained on data sans the influential points from Alpha and Gamma. The differences in RMSE (11.21% and 10.25%) were somewhat significant, as were the differences in adjusted R-squared (9.05% and 8.37%), and the percent of variance explained (8.81% and 7.6%).

Prediction Accuracy

The evaluate the effects of model selection method and the treatment of outliers on prediction accuracy, the four multiregression models were evaluated for prediction accuracy on the test data. Four measures of prediction accuracy were used:

MAPE - Mean Absolute Percentage Error
MPE - Mean Percentage Error
MSE - Mean Squared Error
RMSE - Root Mean Squared Error

In addition, a percent accuracy measure was computed as the percentage of the observations in the test set in which the actual log number of IMDB votes fell within the prediction interval.

Table 24 Model Predictive Accuracy Summary

Model	Size	F Statistic	R-Squared	Adj R-Squared	% Variance	MAPE	MPE	MSE	RMSE	X..Accuracy
Model Alpha	9	29.625	0.642	0.621	64.227	8.415	-1.165	2.069	1.438	94.355
Model Beta	10	31.618	0.699	0.677	69.885	8.372	-1.578	2.156	1.468	93.548
Model Gamma	13	23.956	0.648	0.621	64.823	8.373	-0.894	2.072	1.439	96.774
Model Delta	13	28.660	0.698	0.673	69.752	8.444	-1.374	2.124	1.457	93.548

There were no significant differences in MAPE, MSE, and RMSE between the models. The negative MPE indicates that all models were biased with over predictions. From a percent accuracy perspective, it is worth noting that the forward selection and backward selection models performed identically with and without the influence points. Indeed, the Alpha and Gamma models performed equally well; however, the Alpha model was able to do so with just 4 fewer variables. Therefore, the most parsimonious model, Alpha would advance to the movie prediction stage.

The Model

The prediction equation was defined as follows: $y_i = $ 9.932304 + 0.589$x_1$ + -1.422$x_2$ + -2.225$x_3$ + -0.976$x_4$ + -2.633$x_5$ + -1.508$x_6$ + -0.583$x_7$ + -1.929$x_8$ + -1.039$x_9$ + -1.564$x_10$ + -0.231$x_11$ + 0.014$x_12$ + 1.128$x_13$ + -0.841$x_14$ + 0.676$x_15$ + 0.289$x_16$ + -0.038$x_17$

Movie Rating Model and Predictor